On several partitioning problems of Bollobلs and Scott

Judicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let G be a hypergraph with m i edges of size i for i = 1 , 2 . We show that for any integer k ⩾ 1 , V ( G ) admits a partition into k sets each containing at most m 1 / k + m 2 / k 2 + o ( m 2 ) edges, establishing a conjecture of Bollobás and Scott. We also prove that V ( G ) admits a partition into k ⩾ 3 sets, each meeting at least m 1 / k + m 2 / ( k − 1 ) + o ( m 2 ) edges, which, for large graphs, implies a conjecture of Bollobás and Scott (the conjecture is for all graphs). For k = 2 , we prove that V ( G ) admits a partition into two sets each meeting at least m 1 / 2 + 3 m 2 / 4 + o ( m 2 ) edges, which solves a special case of a more general problem of Bollobás and Scott.